Collaborative Research: New Directions in Nonparametric Inference on Manifolds with Applications to Shapes and Images

3D结构的投影形状，而不是它们的肯德尔型相似形状，是机器视觉和机器人技术中最适合进行一般图像分析的对象。本项目将通过构造完整投影形状流形的适当等变嵌入，并提供基于嵌入下的外在平均值的两样本和多样本推理程序来发展无配准的非参数推理。另一方面，对尺寸和形状反射相似流形的数据分析在蛋白质和人类骨骼结构的各种构型的虚拟重建中是重要的。该项目的另一个重点是某些特殊空间，这些空间不是流形，而是具有流形分层的空间，并且在许多应用中出现，例如在系统发育树的几何表示中。除了如上所述的基于形状分析的地标之外，由二维边界轮廓给出的连续形状将作为无限维（希尔伯特）流形的元素进行研究。最后，提出的用于形状流形密度估计、回归和分类的非参数贝叶斯过程将是迄今为止主要基于Fre'chet平均值和色散的非参数推理的重要出发点。这些项目旨在为形状提供全面可靠的程序，这些程序在许多科学和技术领域都具有广泛的适用性。今天，数字图像在科学技术、情报收集和国防以及日常生活的许多方面发挥着至关重要的作用。本提案旨在通过对形状和其他非欧几里得物体的统计研究来推进对数码相机图像的分析。pi和其他人在过去12年中开发的非参数统计方法对普通数码相机的3D场景识别的统计推断、医学诊断和许多其他形式的图像分析产生了重大影响。该提案的目的不仅是巩固这一理论。目标还包括开发机器视觉和机器人技术的新方法，用于动态场景识别，用于从CT扫描中进行医学诊断，用于为严重受伤的人计划重建手术，以及通过蛋白质的形状配置从DNA序列中检测难以捉摸的健康损伤。

Projective shapes of 3D configurations, not their Kendall type similarity shapes, are the most appropriate objects for general image analysis in machine vision and robotics. The present project will develop registration free nonparametric inference by constructing an appropriate equivariant embedding of the full projective shape manifold, and by providing two-sample and multi-sample inference procedures based on the extrinsic mean under the embedding. On the other hand, data analysis on size-and-shape reflection-similarity manifolds are important in virtual reconstructions of proteins and of various configurations of bone structures in humans. Another focus of the project is on certain special spaces which are not manifolds, but are spaces with manifold stratification, and which arise in many applications, e.g., in geometric representations of phylogenetic trees. Apart from the landmarks based shape analysis as described above, continuous shapes such as given by boundary contours in 2D will be investigated as elements of infinite dimemsional (Hilbert) manifolds. Finally, proposed nonparametric Bayesian procedures for density estimation, regression and classification on shape manifolds will be a significant point of departure from nonparametric inference based so far primarily on Fre'chet means and dispersions. Together these projects aim at providing comprehensive robust procedures for shapes which are of wide applicability in many fields of science and technology.Digital images today play a vital role in science and technology, in intelligence gathering and defense, and in many aspects of everyday life. The present proposal seeks to advance the analysis of digital camera images via the statistical study of shapes and other non-Euclidean objects. Nonparametric statistical methods developed by the PIs and others over the past twelve years have had a significant impact on statistical inference for 3D scene recognition from regular digital cameras, on medical diagnostics, and on many other forms of image analysis. The proposal aims not only to consolidate this theory. The objective is also to develop new methodologies for machine vision and robotics, for dynamic scene recognition, for medical diagnostics from CT scans for planning reconstructive surgery for the severely injured, and for the detection of elusive health impairments from DNA sequences via shape configurations of proteins.